432 Class 22

Thomas E. Love, Ph.D.

2025-04-03

Today’s Topic: Time-to-Event Data

  • Before Spring Break, we discussed
    • Kaplan-Meier Estimation of the Survival Function
    • Creating Survival Objects, Drawing Survival Curves
    • Testing the difference between Survival Curves
  • Today: start Cox Proportional Hazards Regression
    • The Hazard Function and its Estimation

See Chapters 29-31 of our Course Notes

Today’s R Setup

knitr::opts_chunk$set(comment=NA)

library(janitor)
library(naniar)
library(haven)
library(here)
library(conflicted)
library(broom)
library(gt)
library(rms)
library(survival)
library(survminer)
library(easystats)
library(tidyverse)

theme_set(theme_bw())

The Stanford Heart Transplant Study

The Stanford University Heart Transplant Study1 examined whether an experimental heart transplant program increased lifespan. The heart_tr.sav data (saved as an SPSS file) includes 103 observations on 7 variables, as we’ll see.

  • The data include survtime, which is the number of days the subject was alive after the date they were determined to be a candidate for heart transplant until the termination of the study.

Our first step is to ingest the data, and build a tibble.

What’s in the heart_tr data?

Variable Description
id Random last name, assigned to the subject by Dr. Love
age subject’s age in years at the start of the study
survived survival status (alive or dead) at end of follow-up
survtime days subject was alive from the start of the study to its end
prior did the subject have a prior surgery (yes or no)
transplant treatment (received a transplant) or control (did not)
wait waiting time for transplant (in days)

Ingesting data from an SPSS file

We’ll use the read_spss() function1, from the haven package.

heart_tr <- read_spss(here("c22/data/heart_tr.sav"))

dim(heart_tr)
[1] 103   7
names(heart_tr)
[1] "id"         "age"        "survived"   "survtime"   "prior"     
[6] "transplant" "wait"

Looking at the heart_tr data

heart_tr
# A tibble: 103 × 7
   id          age survived  survtime prior     transplant     wait
   <chr>     <dbl> <dbl+lbl>    <dbl> <dbl+lbl> <dbl+lbl>     <dbl>
 1 Akter        51 2 [dead]         6 2 [no]    2 [control]      NA
 2 Alcorn       30 2 [dead]        50 2 [no]    2 [control]      NA
 3 Ali          33 1 [alive]     1799 2 [no]    1 [treatment]    25
 4 Alway        40 2 [dead]        39 2 [no]    1 [treatment]    36
 5 Amadeo       20 2 [dead]        18 2 [no]    2 [control]      NA
 6 Aybar        54 2 [dead]         3 2 [no]    2 [control]      NA
 7 Bargiel      50 2 [dead]       675 2 [no]    1 [treatment]    51
 8 Bayse        45 2 [dead]        40 2 [no]    2 [control]      NA
 9 Beason       47 2 [dead]        85 2 [no]    2 [control]      NA
10 Boddicker    42 2 [dead]        58 2 [no]    1 [treatment]    12
# ℹ 93 more rows
  • Note the <dbl+lbl> types for the columns containing categorical information.

What are the labels for survived?

  • Here are two options for figuring this out:
heart_tr |> count(survived) # numeric codes, with labels
# A tibble: 2 × 2
  survived      n
  <dbl+lbl> <int>
1 1 [alive]    28
2 2 [dead]     75
print_labels(heart_tr$survived) 

Labels:
 value label
     1 alive
     2  dead
  • Suppose we want to use a factor, without these specialized value labels, to represent the information in survived.

Converting survived into a factor, without labels

heart_tr <- heart_tr |>
  mutate(survived = fct_recode(factor(survived), 
                               "alive" = "1", "dead" = "2"))

heart_tr |> count(survived) # now a factor, without labels
# A tibble: 2 × 2
  survived     n
  <fct>    <int>
1 alive       28
2 dead        75

Now, we have a factor representation of the survived information, with values that make sense.

Converting prior into a factor

heart_tr |> count(prior) # numeric codes, with labels
# A tibble: 2 × 2
  prior         n
  <dbl+lbl> <int>
1 1 [yes]      12
2 2 [no]       91
heart_tr <- heart_tr |>
  mutate(prior = fct_recode(factor(prior), "yes" = "1", "no" = "2"))
  • and now, we have a meaningful factor for prior, too.
heart_tr |> count(prior) # now a factor, without labels
# A tibble: 2 × 2
  prior     n
  <fct> <int>
1 yes      12
2 no       91

tranplant into a 0/1 indicator

Instead of making transplant into a factor, we’ll create a numeric description of the transplant information, where transplant = 1 will mean that the subject did have a heart transplant, and transplant = 0 will mean that the subject did not have a heart transplant.

heart_tr |> count(transplant) # numeric codes, with labels
# A tibble: 2 × 2
  transplant        n
  <dbl+lbl>     <int>
1 1 [treatment]    69
2 2 [control]      34

Converting transplant to 0/1

heart_tr <- heart_tr |>
  mutate(transplant = as.numeric(transplant == 1))

heart_tr |> count(transplant) # 1/0, no labels
# A tibble: 2 × 2
  transplant     n
       <dbl> <int>
1          0    34
2          1    69

heart_tr after our changes

glimpse(heart_tr)
Rows: 103
Columns: 7
$ id         <chr> "Akter", "Alcorn", "Ali", "Alway", "Amadeo", "Aybar", "Barg…
$ age        <dbl> 51, 30, 33, 40, 20, 54, 50, 45, 47, 42, 47, 53, 54, 53, 53,…
$ survived   <fct> dead, dead, alive, dead, dead, dead, dead, dead, dead, dead…
$ survtime   <dbl> 6, 50, 1799, 39, 18, 3, 675, 40, 85, 58, 153, 8, 81, 1386, …
$ prior      <fct> no, no, no, no, no, no, no, no, no, no, no, no, no, no, no,…
$ transplant <dbl> 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1,…
$ wait       <dbl> NA, NA, 25, 36, NA, NA, 51, NA, NA, 12, 26, NA, 17, 37, NA,…
  • So we no longer have any labels, and our categorical variables are presented as factors (survived and prior) or as a 1/0 numeric variable (transplant.)
  • Because our subject identifier id was a set of (fake) last names, rather than numbers, this is already a character variable, which is what we want. Are they unique?
identical(nrow(heart_tr), n_distinct(heart_tr$id))
[1] TRUE

Creating a Survival Object

  • survtime shows the in-study time (in days) until death or censoring
  • survived is a factor showing “dead” or “alive”.
heart_tr$S <- Surv(time = heart_tr$survtime, 
                   event = heart_tr$survived == "dead")

head(heart_tr$S)
[1]    6    50  1799+   39    18     3
  • The first subject died after 6 days, while the second died after 50 days. The third subject was censored after 1799 days.

Compare the two transplant groups

km_tr <- survfit(S ~ transplant, data = heart_tr)

km_tr
Call: survfit(formula = S ~ transplant, data = heart_tr)

              n events median 0.95LCL 0.95UCL
transplant=0 34     30     21      12      50
transplant=1 69     45    285     153     852

Plotting the K-M Curves

ggsurvplot(km_tr, data = heart_tr, palette = "aaas")

Kaplan-Meier Estimates (table)

summary(km_tr)
Call: survfit(formula = S ~ transplant, data = heart_tr)

                transplant=0 
 time n.risk n.event survival std.err lower 95% CI upper 95% CI
    1     34       1   0.9706  0.0290       0.9154        1.000
    2     33       3   0.8824  0.0553       0.7804        0.998
    3     30       3   0.7941  0.0693       0.6692        0.942
    5     27       1   0.7647  0.0727       0.6346        0.921
    6     26       2   0.7059  0.0781       0.5682        0.877
    8     24       1   0.6765  0.0802       0.5362        0.853
    9     23       1   0.6471  0.0820       0.5048        0.829
   12     21       1   0.6162  0.0836       0.4723        0.804
   16     20       1   0.5854  0.0849       0.4405        0.778
   18     19       1   0.5546  0.0859       0.4094        0.751
   21     18       2   0.4930  0.0867       0.3493        0.696
   32     15       1   0.4601  0.0869       0.3177        0.666
   35     14       1   0.4273  0.0867       0.2871        0.636
   36     13       1   0.3944  0.0860       0.2572        0.605
   37     12       1   0.3615  0.0849       0.2281        0.573
   40     11       2   0.2958  0.0812       0.1727        0.507
   50      9       1   0.2629  0.0786       0.1464        0.472
   69      8       1   0.2301  0.0753       0.1211        0.437
   85      7       1   0.1972  0.0714       0.0970        0.401
  102      6       1   0.1643  0.0666       0.0743        0.364
  149      5       1   0.1315  0.0609       0.0531        0.326
  263      4       1   0.0986  0.0538       0.0338        0.287
  340      3       1   0.0657  0.0448       0.0173        0.250

                transplant=1 
 time n.risk n.event survival std.err lower 95% CI upper 95% CI
    5     69       1    0.986  0.0144       0.9577        1.000
   16     68       2    0.957  0.0246       0.9096        1.000
   17     66       1    0.942  0.0281       0.8885        0.999
   28     65       1    0.928  0.0312       0.8683        0.991
   30     64       1    0.913  0.0339       0.8489        0.982
   39     63       1    0.899  0.0363       0.8301        0.973
   43     61       1    0.884  0.0386       0.8113        0.963
   45     60       1    0.869  0.0407       0.7929        0.953
   51     59       1    0.854  0.0426       0.7748        0.942
   53     58       1    0.840  0.0443       0.7571        0.931
   58     57       1    0.825  0.0459       0.7396        0.920
   61     56       1    0.810  0.0474       0.7224        0.909
   66     55       1    0.795  0.0488       0.7053        0.897
   68     54       2    0.766  0.0512       0.6719        0.873
   72     52       2    0.737  0.0533       0.6391        0.849
   77     50       1    0.722  0.0543       0.6229        0.836
   78     49       1    0.707  0.0551       0.6069        0.824
   80     48       1    0.692  0.0559       0.5910        0.811
   81     47       1    0.678  0.0566       0.5752        0.798
   90     46       1    0.663  0.0573       0.5596        0.785
   96     45       1    0.648  0.0579       0.5441        0.772
  100     44       1    0.633  0.0584       0.5287        0.759
  110     42       1    0.618  0.0589       0.5130        0.745
  153     40       1    0.603  0.0594       0.4969        0.731
  165     39       1    0.587  0.0599       0.4810        0.717
  186     37       1    0.572  0.0603       0.4647        0.703
  188     36       1    0.556  0.0607       0.4485        0.688
  207     35       1    0.540  0.0610       0.4325        0.674
  219     34       1    0.524  0.0613       0.4166        0.659
  285     32       2    0.491  0.0616       0.3840        0.628
  308     30       1    0.475  0.0617       0.3680        0.613
  334     29       1    0.458  0.0617       0.3521        0.597
  342     27       1    0.441  0.0617       0.3356        0.581
  583     20       1    0.419  0.0625       0.3132        0.562
  675     16       1    0.393  0.0638       0.2860        0.540
  733     15       1    0.367  0.0647       0.2597        0.519
  852     13       1    0.339  0.0656       0.2317        0.495
  979     10       1    0.305  0.0672       0.1979        0.470
  995      9       1    0.271  0.0678       0.1660        0.442
 1032      8       1    0.237  0.0672       0.1360        0.413
 1386      5       1    0.190  0.0685       0.0935        0.385

How about a log rank test?

survdiff(S ~ transplant, data = heart_tr)
Call:
survdiff(formula = S ~ transplant, data = heart_tr)

              N Observed Expected (O-E)^2/E (O-E)^2/V
transplant=0 34       30     12.1      26.5      33.2
transplant=1 69       45     62.9       5.1      33.2

 Chisq= 33.2  on 1 degrees of freedom, p= 8e-09
  • What can we conclude from this result?

Log-Log Plot for K-M estimation

  • The two curves do not meet during the observation period, indicating the satisfaction of the proportional hazard assumption made by the log rank test.
plot(survfit(S ~ transplant, data = heart_tr), col = c(1,2), fun = "cloglog")

Cumulative Event Rate for km_tr

  • Add fun = “event” to our ggsurvplot(), and
  • A table of subjects at risk over time, and
  • The p value from the log rank test.
ggsurvplot(km_tr, data = heart_tr, palette = "aaas",
           fun = "event",
           xlab = "Survival time in days", 
           break.time.by = 365,
           risk.table = TRUE, risk.table.height = 0.25,
           pval = TRUE, pval.method = TRUE, 
           pval.size = 4, pval.method.size = 4,
           pval.coord = c(900, 0.20), pval.method.coord = c(900, 0.30))

Cumulative Event Rate for km_tr

The Hazard Function H(t)

If S(t) is the survival function, and time t is taken to be continuous, then the hazard function H(t) is defined as:

\[ S(t) = e^{H(t)} \mbox{ and, thus, } H(t) = -ln(S(t)) \]

  • H(t) is used to describe the concept of the risk of “failure” in an interval after time t, conditioned on the subject having survived to time t.
  • H(t) is the cumulative hazard function, to emphasize that its value is the “sum” of the hazard up to time t.

Understaning the Hazard Function

Consider a subject in the heart transplant study who has a survival time of 1000 days. Let’s ignore the transplant group information for a moment.

  • For this subject to die at 1000 days, they had to survive for the first 999 days.
  • The subject’s hazard at 1000 days is the failure rate “per day” conditional on the subject being alive through the first 999 days.

Estimating the Hazard Function

Suppose we want to estimate H(t) across all subjects.

  • There are several different methods, but we’ll focus on the inverse Kaplan-Meier estimator.

I’ll build something called H.est1, the inverse K-M estimate…

km_1 <- survfit(S ~ 1, data = heart_tr)

Haz1.almost <- -log(km_1$surv)

H_est1 <- c(Haz1.almost, tail(Haz1.almost, 1))

Tibble of times and hazard estimates

haz_tib <- tibble(
  time = c(km_1$time, tail(km_1$time, 1)),
  inverse_KM = H_est1)

haz_tib
# A tibble: 89 × 2
    time inverse_KM
   <dbl>      <dbl>
 1     1    0.00976
 2     2    0.0396 
 3     3    0.0704 
 4     5    0.0914 
 5     6    0.113  
 6     8    0.124  
 7     9    0.135  
 8    11    0.135  
 9    12    0.146  
10    16    0.181  
# ℹ 79 more rows

Cumulative Hazard Function from Inverse Kaplan-Meier Estimates

ggplot(haz_tib, aes(x = time, y = inverse_KM)) + 
    geom_step() + 
    scale_x_continuous(breaks = c(0, 365, 730, 1095, 1460, 1825)) +
    labs(x = "Days of Follow-Up", 
         y = "Cumulative Hazard of Death",
         title = "Cumulative Hazard Function via Inverse K-M")

Cumulative Hazard Function from Inverse Kaplan-Meier Estimates

Cumulative Hazard Function from Inverse Kaplan-Meier Estimates

ggsurvplot(km_1, data = heart_tr, fun = "cumhaz", conf.int = FALSE,
           xlab = "Days of Follow-Up", break.time.by = 365)

Plotting Cumulative Hazard by Transplant Group

For our km_tr fit, we’d use

ggsurvplot(km_tr, data = heart_tr, fun = "cumhaz",
           xlab = "Days of Follow-Up", palette = "aaas",
           pval = TRUE, pval.coord = c(1450, 0.2),
           break.time.by = 365,
           risk.table = TRUE, risk.table.height = 0.25)

Plotting Cumulative Hazard by Transplant Group

Cox Proportional Hazards Regression

fit1_tr <- coxph(S ~ transplant, data = heart_tr)

The Cox proportional hazards model fits survival data with a constant (not varying over time) covariate (here, transplant group) to a hazard function of the form:

\[ h(t|transplant) = h_0(t)exp(\beta_1 transplant) \]

where we estimate the unknown value of \(\beta_1\) and where \(h_0(t)\) is the baseline hazard which depends on time \(t\) but not on the transplant group.

Cox Model fit1_tr

fit1_tr
Call:
coxph(formula = S ~ transplant, data = heart_tr)

              coef exp(coef) se(coef)      z        p
transplant -1.3234    0.2662   0.2438 -5.428 5.69e-08

Likelihood ratio test=25.95  on 1 df, p=3.511e-07
n= 103, number of events= 75

Our hazard ratio estimate is 0.2662 for transplant group 1 (vs. transplant group 0)

  • Hazard ratio < 1 indicates a decrease in hazard for subjects who received a transplant as compared to those who did not. Does this match our plot (repeated on next slide)?

Plotting Cumulative Hazard by Transplant Group

Cox Model Parameters (fit1_tr)

model_parameters(fit1_tr, pretty_names = FALSE, ci = 0.90, digits = 3)
Parameter  | Coefficient |    SE |           90% CI |      z |      p
---------------------------------------------------------------------
transplant |      -1.323 | 0.244 | [-1.724, -0.922] | -5.428 | < .001
model_parameters(fit1_tr, pretty_names = FALSE, ci = 0.90, digits = 3,
                 exponentiate = TRUE)
Parameter  | Coefficient |    SE |         90% CI |      z |      p
-------------------------------------------------------------------
transplant |       0.266 | 0.065 | [0.178, 0.398] | -5.428 | < .001
  • Compare to tidy() results from broom?
tidy(fit1_tr, exponentiate = TRUE, conf.int = TRUE, conf.level = 0.90) |>
  gt() |> tab_options(table.font.size = 20) |> 
  fmt_number(decimals = 3) |> opt_stylize(style = 4, color = "blue")
term estimate std.error statistic p.value conf.low conf.high
transplant 0.266 0.244 −5.428 0.000 0.178 0.398

Cox Model fit1_tr Performance

model_performance(fit1_tr)
Response residuals not available to calculate mean square error. (R)MSE
  is probably not reliable.
Warning: Can't calculate weighted residuals from model.
# Indices of model performance

AIC     |    AICc |     BIC | Nagelkerke's R2 |  RMSE | Sigma
-------------------------------------------------------------
572.297 | 572.336 | 574.931 |           0.223 | 0.947 | 0.000
  • How about glance() from broom?
glance(fit1_tr) |> gt() |> fmt_number(decimals = 2)
n nevent statistic.log p.value.log statistic.sc p.value.sc statistic.wald p.value.wald statistic.robust p.value.robust r.squared r.squared.max concordance std.error.concordance logLik AIC BIC nobs
103.00 75.00 25.95 0.00 33.38 0.00 29.47 0.00 NA NA 0.22 1.00 0.67 0.02 −285.15 572.30 574.61 75.00

Forest Plot for fit1_tr model

ggforest(fit1_tr)

Plot Adjusted Survival Curves

ggadjustedcurves(fit1_tr, data = data.frame(heart_tr), 
                 method = "average", variable = "transplant")

Checking the Proportional Hazards Assumption

If the proportional hazards assumption is appropriate, we should see a slope of essentially zero in the residuals that are plotted against time on the next slide.

  • If we see a slope that seriously different from zero, that will suggest a violation of the proportional hazards assumption.
  • A hypothesis test is also performed, where a small p value indicates a potential problem with the assumption.

Plot to Check Proportional Hazards

ggcoxzph(cox.zph(fit1_tr))

Cox Model Diagnostics (fit1_tr)

ggcoxdiagnostics(fit1_tr)

What if we also include age?

fit2_tr <- coxph(S ~ transplant + age, data = heart_tr)

fit2_tr
Call:
coxph(formula = S ~ transplant + age, data = heart_tr)

               coef exp(coef) se(coef)      z        p
transplant -1.78994   0.16697  0.27107 -6.603 4.02e-11
age         0.06020   1.06205  0.01527  3.943 8.03e-05

Likelihood ratio test=44.55  on 2 df, p=2.121e-10
n= 103, number of events= 75

fit2_tr model parameters

model_parameters(fit2_tr, exponentiate = TRUE, pretty_names = FALSE, 
                 ci = 0.90, digits = 3)
Parameter  | Coefficient |    SE |         90% CI |      z |      p
-------------------------------------------------------------------
transplant |       0.167 | 0.045 | [0.107, 0.261] | -6.603 | < .001
age        |       1.062 | 0.016 | [1.036, 1.089] |  3.943 | < .001
  • If Harry is a year older than Steve and each are in the same transplant group, then Harry’s hazard of death is 1.062 (90% CI 1.036, 1.089) times that of Steve.
  • If Harry and Sally are the same age, but Sally received a transplant but Harry did not, then Sally’s hazard of death is 0.167 (90% CI 0.107, 0.261) times that of Harry.

fit2_tr R-square measures

model_performance(fit2_tr)
# Indices of model performance

AIC     |    AICc |     BIC | Nagelkerke's R2 |  RMSE | Sigma
-------------------------------------------------------------
555.695 | 555.815 | 560.965 |           0.352 | 0.880 | 0.000
  • model_parameters() gives the Nagelkerke \(R^2\).
glance(fit2_tr) |> select(n, nevent, nobs, r.squared, r.squared.max) |>
  gt() |> fmt_number(columns = r.squared:r.squared.max, decimals = 3) |>
  tab_options(table.font.size = 20) |> opt_stylize(style = 5, color = "blue")
n nevent nobs r.squared r.squared.max
103 75 75 0.351 0.997
  • glance() gives the Cox-Snell \(R^2\) along with its maximum value (< 1.)

fit2_tr concordance measure

glance(fit2_tr) |> select(n, nevent, nobs, concordance, 
                          se_conc = std.error.concordance) |>
  gt() |> fmt_number(columns = concordance:se_conc, decimals = 3) |>
  tab_options(table.font.size = 20) |> opt_stylize(style = 5, color = "blue")
n nevent nobs concordance se_conc
103 75 75 0.721 0.034

Compare the model’s prediction for a pair of observations in the data. A pair is concordant if the prediction and data agree in direction. Concordance is the fraction of pairs that are concordant.

  • Higher Concordance = better Cox model predictions.

Forest Plot for fit2_tr model

ggforest(fit2_tr)

Plot Adjusted Survival Curves

ggadjustedcurves(fit2_tr, data = data.frame(heart_tr),
                 method = "average", variable = "transplant")

Effect of Adding age?

  • fit1_tr includes transplant group, fit2_tr adds age.
plot(compare_performance(fit1_tr, fit2_tr))

ANOVA comparing fit1_tr to fit2_tr

anova(fit1_tr, fit2_tr)
Analysis of Deviance Table
 Cox model: response is  S
 Model 1: ~ transplant
 Model 2: ~ transplant + age
   loglik  Chisq Df Pr(>|Chi|)    
1 -285.15                         
2 -275.85 18.602  1  1.611e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Plot to Check Proportional Hazards

ggcoxzph(cox.zph(fit2_tr))

Cox Model Diagnostics (fit2_tr)

ggcoxdiagnostics(fit2_tr)

Model fit3_tr adding prior surgery

fit3_tr <- coxph(S ~ transplant + age + prior, data = heart_tr)

fit3_tr
Call:
coxph(formula = S ~ transplant + age + prior, data = heart_tr)

               coef exp(coef) se(coef)      z        p
transplant -1.66121   0.18991  0.27588 -6.021 1.73e-09
age         0.05919   1.06098  0.01494  3.962 7.44e-05
priorno     0.74266   2.10152  0.44225  1.679   0.0931

Likelihood ratio test=47.89  on 3 df, p=2.247e-10
n= 103, number of events= 75

fit3_tr model parameters

model_parameters(fit3_tr, exponentiate = TRUE, pretty_names = FALSE, 
                 ci = 0.90, digits = 3)
Parameter  | Coefficient |    SE |         90% CI |      z |      p
-------------------------------------------------------------------
transplant |       0.190 | 0.052 | [0.121, 0.299] | -6.021 | < .001
age        |       1.061 | 0.016 | [1.035, 1.087] |  3.962 | < .001
priorno    |       2.102 | 0.929 | [1.015, 4.350] |  1.679 | 0.093

fit3_tr R-square measures

model_performance(fit3_tr)
# Indices of model performance

AIC     |    AICc |     BIC | Nagelkerke's R2 |  RMSE | Sigma
-------------------------------------------------------------
554.352 | 554.595 | 562.256 |           0.373 | 0.893 | 0.000
glance(fit3_tr) |> select(n, nevent, nobs, r.squared, r.squared.max) |>
  gt() |> fmt_number(columns = r.squared:r.squared.max, decimals = 3) |>
  tab_options(table.font.size = 20) |> opt_stylize(style = 5, color = "blue")
n nevent nobs r.squared r.squared.max
103 75 75 0.372 0.997

fit3_tr concordance

glance(fit3_tr) |> select(n, nevent, nobs, concordance, 
                          se_conc = std.error.concordance) |>
  gt() |> fmt_number(columns = concordance:se_conc, decimals = 3) |>
  tab_options(table.font.size = 20) |> opt_stylize(style = 5, color = "blue")
n nevent nobs concordance se_conc
103 75 75 0.739 0.031
  • Assesses probability of agreement between survival time and the risk score generated by the predictors
  • 1 indicates perfect agreement, 0.5 indicates no better than chance

Compare our 3 models

plot(compare_performance(fit1_tr, fit2_tr, fit3_tr))

Checking PH Assumption for fit3_tr

ggcoxzph(cox.zph(fit3_tr))

Cox Model Diagnostics for fit3_tr

ggcoxdiagnostics(fit3_tr)

What happens if we see a violation?

  • We could add a non-linear predictor term or use a different kind of survival model.

  • If the PH assumption fails on a categorical predictor, fit a Cox model stratified by that predictor (use strata(var) rather than var in the specification of the coxph model.)

  • If the PH assumption is violated, this means the hazard isn’t constant over time, so we could fit separate Cox models for a series of time intervals.

If we see a violation…

Another option would be to use an extension of the Cox model that permits covariates to vary over time.

Visit https://cran.r-project.org/web/packages/survival/vignettes/timedep.pdf for details on building the relevant data sets and models, with examples.

432 Class 22 | 2025-04-03 | https://thomaselove.github.io/432-2025/